Proof: Thales’ Theorem

In this post I’ll be demonstrating how you can prove that Thales’ Theorem is true. To follow the steps in this post (11 in total), what you will require is a ruler, pair of compasses and a pencil.

Step 1: Draw a random line on a sheet of paper.

Random line....

Step 2: Place your compass needle on this line, and form a circle.

A circle...

Step 3: Add 4 points to your drawing, as shown below…

Add 4 points to your drawing....

Step 4: Name the points A, B, C and D as shown…

Points A, B, C and D

Step 5: Connect the points A, B and D together to form an isosceles triangle

Points A, B and D connected.

Step 6: Name the lines AB and BD the radius (r)…

Label the radiuses...

Step 7: Since the lines AB and BD are equal to one another, it follows that the angles ∠BAD and ∠BDA are equivalent. This is because the angles below the apex of an isosceles triangle are equal. You must name these angles alpha (α).

Label alpha angles...

Step 8: Now connect the points BC and CD together to form another isosceles triangle…

Another isosceles triangle...

Step 9: The line BC is equal to r… Now label the line BC…

Label the line BC...

Step 10: Because the line BC and BD are both equal to r, the triangle BCD is an isosceles triangle. This means that the angles ∠BCD and ∠BDC must both be equivalent. Call these angles beta (β).

Label the angles beta...

Step 11: Prove that the angle at point D is equal to 90 degrees.

Thales’ Theorem is as follows:

Because AC is the diameter of the circle you drew, the angle at the point D (α+β) must be equal to 90 degrees. In more specific and general terms, if you have the points A, C and D lying on a circle – and the line AC is in fact the diameter of this circle – then the angle at point D (α+β) must be a right angle.

Proof (which must be derived using the diagram you’ve created):

Diagram.

All angles within a triangle (in 2 space) must add up to 180 degrees.

Mathematically, this means that:

\alpha +\alpha +\beta +\beta =180\\ \\ \Rightarrow \quad 2\alpha +2\beta =180\\ \\ \Rightarrow \quad 2\left( \alpha +\beta  \right) =180\\ \\ \Rightarrow \quad \frac { 2\left( \alpha +\beta  \right)  }{ 2 } =\frac { 180 }{ 2 } \\ \\ \therefore \quad \alpha +\beta =90

And as a result, Thales’ theorem must be true. The angle α+β is the angle at point D.

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